    
We turn to the question about the reality of the Lorentz contraction of a Dirac wave function. In order to be clear let us first recall briefly some well-known general facts (see e.g. [1]}). Let \(\mathfrak {R}\)  denote the standard basis \((\mathfrak {e}^0,\mathfrak {e}^1,\mathfrak {e}^2,\mathfrak {e}^3)\)  of \(\mathbb {R}^4\)  as well the reference frame related to it.
The description of the physical system will refer to \(\mathfrak {R}\)  in determining the representation of its states \(S\)  and observables \(O\)  by unit vectors \(\psi \)  and self-adjoint operators \(A\) , respectively, in the Hilbert space of states. The outcomes of measurements are the expectation values \(\langle \psi , A \psi \rangle \) . The state space is unique up to Hilbert space isomorphisms \(\iota \) , as \(\iota \psi \)  and \(\iota A \iota ^{-1}\)  yield the same expectation values as \(\psi \)  and \(A\) , i.e., \( \langle \iota \psi , (\iota A \iota ^{-1})\,\iota \psi \rangle = \langle \psi , A \psi \rangle \) . So every \(\iota \)  determines another representation of the states \(S\)  and observables \(O\)  of the system with \(\iota \psi \)  and \(\iota A \iota ^{-1}\)  denoting the same state and observable, respectively, as \(\psi \)  and \(A\)  do. This freedom plays a decisive role in the description of symmetries. In particular relativistic symmetry gives rise to a representation \(W\)  of \(\tilde{\mathcal {P}}\)  in the state space.
